time-dependent harmonic oscillators TimeDependentHarmonicOscillators 2009-05-29 13:46:17 2009-05-29 14:38:12 Topic Ermakov systems superposition function Lie's geometric approach non-autonomous systems of first-order DEs exact analytic Gaussian wave packet (WP) solutions non-linear equations nonlinear equations second-order differential equations Milne--Pinney equation Ermakov--Lewis invariants Riccati equation time-dependent harmonic oscillators % this is the default PlanetPhysics preamble. as your \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate} \usepackage{xypic, xspace} \usepackage[mathscr]{eucal} \usepackage[dvips]{graphicx} \usepackage[curve]{xy} % define commands here \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\grpL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\rO}{{\rm O}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\SL}{{\rm Sl}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\Symb}{{\rm Symb}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} %\newcommand{\grp}{\mathcal G} \renewcommand{\H}{\mathcal H} \renewcommand{\cL}{\mathcal L} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathsf{G}}} \newcommand{\dgrp}{{\mathsf{D}}} \newcommand{\desp}{{\mathsf{D}^{\rm{es}}}} \newcommand{\grpeod}{{\rm Geod}} %\newcommand{\grpeod}{{\rm geod}} \newcommand{\hgr}{{\mathsf{H}}} \newcommand{\mgr}{{\mathsf{M}}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathsf{G)}}} \newcommand{\obgp}{{\rm Ob(\mathsf{G}')}} \newcommand{\obh}{{\rm Ob(\mathsf{H})}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\grphomotop}{{\rho_2^{\square}}} \newcommand{\grpcalp}{{\mathsf{G}(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\grplob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\grpa}{\grpamma} %\newcommand{\grpa}{\grpamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\ovset}[1]{\overset {#1}{\ra}} \newcommand{\ovsetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \newcommand{\<}{{\langle}} %\newcommand{\>}{{\rangle}} %\usepackage{geometry, amsmath,amssymb,latexsym,enumerate} %\usepackage{xypic} \def\baselinestretch{1.1} \hyphenation{prod-ucts} %\grpeometry{textwidth= 16 cm, textheight=21 cm} \newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}& #3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram \eqno{\mbox{#9}}$$ } \def\C{C^{\ast}} \newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}} %\newenvironment{proof}{\noindent {\bf Proof} }{ \hfill $\Box$ %{\mbox{}} \newcommand{\quadr}[4] {\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4& \end{pmatrix}} \def\D{\mathsf{D}} \section{Time-dependent harmonic oscillators} Nonlinear equations are of increasing interest in Physics; Riccati and Ermakov systems enter the formalism of quantum theory in the study of cases where exact analytic Gaussian wave packet (WP) solutions of the time-dependent Schr\"odinger equation (SE) do exist, and in particular, in the harmonic oscillator (HO) and the free motion cases. One of the simplest examples of such nonlinear equations is the Milne--Pinney equation: $$d^2x/dt^2 = -- {\omega}^2(t)x +k x^3,$$ (1) where $k$ is a real constant with values depending on the field in which the equation is to be applied. \subsection{Ermakov systems} This equation was introduced in the nineteenth century by V.P. Ermakov, as a way of looking for a first integral for the time-dependent harmonic oscillator. He employed some of Lie's ideas for dealing with ordinary differential equations with the tools of classical geometry. Lie had previously obtained a characterization of non-autonomous systems of first-order differential equations admitting a superposition rule: $$dx_i/dt = Y i(t, x), i = 1, . . . , n, $$ (2). This approach has been recently reformulated from a geometric perspective in which the role of the superposition function is played by an appropriate algebriac connection. This geometric approach allows one to consider a superposition of solutions of a given system in order to obtain solutions of another system as a kind of mathematical construction that might be generalized even further; such a superposition rule may be understood from a geometric viewpoint in some interesting cases, as in the Milne--Pinney equation (1), or in the Ermakov system and its generalisations. One recalls here that {\em Ermakov systems} are defined as systems of second-order differential equations composed by the Milne--Pinney differential equation (1) together with the corresponding time--dependent harmonic oscillator. Ermakov systems have been also broadly studied in Physics since their introduction in the nineteenth century. They also appear in the study of the Bose--Einstein condensates, cosmological models, and the solution of time--dependent harmonic or anharmonic oscillators. Several recent reports are concerned with the use Hamiltonian or Lagrangian structures in the study of such a system, and many generalisations or new insights from the mathematical point of view have ben thus obtained. {\em Ermakov--Lewis invariants} naturally emerge as functions defining the foliation associated to the superposition rule. It has been shown by Ermakov in 1880 that the system of differential equations coupled via the possibly time-dependent frequency $\omega$, leads to a dynamical invariant that has been rediscovered by several authors in the 20th century: $$I_L = 0.5 (d \eta/dt ~ \alpha~ −~ \eta d \alpha/dt)^2 + (\eta~ \alpha)^2 = const$$. (3) It is straightforward to show that $d/dt (I_L) = 0.$ The above Ermakov invariant $I_L$ depends not only on the classical variables $\eta$ η(t) and its time derivative, but also on the quantum uncertainty related to $\alpha (t)$ and its time derivative. Additional interesting insight into the relation between variables $\eta$ and $\alpha$ can be obtained by considering also the \PMlinkexternal{Riccati equation}{http://en.wikipedia.org/wiki/Riccati_equation}.